Chaos in a closed Friedmann-Robertson-Walker universe: An imaginary approach

Abstract

In this work we study the existence of mechanisms of the transition to global chaos in a closed Friedmann-RobertsonWalker universe with a massive conformally coupled scalar field. We propose a complexification of the radius of the universe so that the global dynamics can be understood. We show numerically the existence of heteroclinic connections of the unstable and stable manifolds to periodic orbits associated with the saddle-center equilibrium points. We find two bifurcations which are crucial in creating non-collapsing universes both in the real version and in the imaginary extension of the models. The techniques presented here can be employed in any cosmological model.

On the integrability of Friedmann–Robertson–Walker models with conformally coupled massive scalar fields

“…From our analysis we have shown rigorously using analytic methods that FRW universes with a conformally coupled massive scalar field are nonintegrable. This is compatible with results from numerical analysis based on Poincaré sections [6] which indicate that the behaviour of the system is mathematically chaotic.…”

“…Kovacic's algorithm can be divided into three main steps: the first step is the determination of the subset of L relevant for the LODE under consideration; the two other steps are devoted respectively to determining the existence of the minimal polynomial, and its construction. If the algorithm does not terminate successfully (ie, equation (6) has no algebraic solution) then equation (4) has no solution in terms of Liouvillian functions.…”

Free motion around black holes with discs or rings: between integrability and chaos - I

“…The latter has naturally been examined in cosmological solutions, mainly in Bianchi IX (Mixmaster) models. While the secular debate about chaoticity and the very nature of these universes continues (Coley 2002; Fay & Lehner 2004; Benini & Montani 2004; Soares & Stuchi 2005; Heinzle, Röhr & Uggla 2006; Buzzi, Llibre & da Silva 2007; Andriopoulos & Leach 2008; Heinzle & Uggla 2009), one of the later attractors is to assess the effect of the cosmological constant and/versus that of a scalar field within the Friedmann–Lematre–Robertson–Walker model dynamics (Jorás & Stuchi 2003; Faraoni, Jensen & Theuerkauf 2006; Hrycyna & Szydłowski 2006; Lukes‐Gerakopoulos, Basilakos & Contopoulos 2008; Maciejewski et al 2008). [We only give several more recent references here.…”

Homoclinic chaos in axisymmetric Bianchi-IX cosmological models with anad hocquantum potential

“…Cornish and Levin [11] proposed to quantify chaos in the Mixmaster universe by calculating the dimensions of fractal basin boundaries in initial-conditions sets for the full dynamics, these boundaries being defined by a code association with one of the three outcomes on which one of the three axes is collapsing most quickly, as established numerically. Jorás and Stuchi [12] examined chaos in FRW models with a coupled scalar field by extending the analysis to the complex plane and found a family of non-collapsing structures. In particular homoclinic * Electronic address: gcoelho@if.ufrj.br † Electronic address: tstuchi@if.ufrj.br ‡ Electronic address: joras@if.ufrj.br chaos in axisymmetric Bianchi IX universes with matter and cosmological constant has been treated in [13] using coordinate independent topological structure of the dynamics to characterize chaotic behavior in the Hamiltonian system and its physical implications.…”